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Geometry and Dynamics in Gromov Hyperbolic Metric Spaces with an Emphasis on Non-Proper Settings Mathematical Surveys and Monographs Volume 218 Geometry and Dynamics in Gromov Hyperbolic Metric Spaces With an Emphasis on Non-Proper Settings Tushar Das David Simmons Mariusz Urbaƙski American Mathematical Society 10.1090/surv/218 Geometry and Dynamics in Gromov Hyperbolic Metric Spaces With an Emphasis on Non-Proper Settings Mathematical Surveys and Monographs Volume 218 Geometry and Dynamics in Gromov Hyperbolic Metric Spaces With an Emphasis on Non-Proper Settings Tushar Das David Simmons Mariusz Urbaƙski American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 20H10, 28A78, 37F35, 20F67, 20E08; Secondary 37A45, 22E65, 20M20. For additional information and updates on this book, visit www.ams.org/bookpages/surv-218 Library of Congress Cataloging-in-Publication Data Names: Das, Tushar, 1980- | Simmons, David, 1988- | Urba´nski, Mariusz. Title: Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non- proper settings / Tushar Das, David Simmons, Mariusz Urba´nski. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Mathe- matical surveys and monographs ; volume 218 | Includes bibliographical references and index. Identifiers: LCCN 2016034629 | ISBN 9781470434656 (alk. paper) Subjects: LCSH: Geometry, Hyperbolic. | Hyperbolic spaces. | Metric spaces. | AMS: Group theory and generalizations – Other groups of matrices – Fuchsian groups and their general- izations. msc | Measure and integration – Classical measure theory – Hausdorff and packing measures. msc | Dynamical systems and ergodic theory – Complex dynamical systems – Con- formal densities and Hausdorff dimension. msc | Group theory and generalizations – Special aspects of infinite or finite groups – Hyperbolic groups and nonpositively curved groups. msc | Group theory and generalizations – Structure and classification of infinite or finite groups – Groups acting on trees. msc | Dynamical systems and ergodic theory – Ergodic theory – Relations with number theory and harmonic analysis. msc | Topological groups, Lie groups – Lie groups – Infinite-dimensional Lie groups and their Lie algebras: general properties. msc | Group theory and generalizations – Semigroups – Semigroups of transformations, etc.. msc Classification: LCC QA685 .D238 2017 | DDC 516.3/62–dc23 LC record available at https://lccn. loc.gov/2016034629 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the authors. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Dedicated to the memory of our friend Bernd O. Stratmann Mathematiker 17th July 1957 – 8th August 2015 Contents List of Figures xi Prologue xiii Chapter 1. Introduction and Overview xvii 1.1. Preliminaries xviii 1.1.1. Algebraic hyperbolic spaces xviii 1.1.2. Gromov hyperbolic metric spaces xviii 1.1.3. Discreteness xxi 1.1.4. The classification of semigroups xxii 1.1.5. Limit sets xxiii 1.2. The Bishop–Jones theorem and its generalization xxiv 1.2.1. The modified Poincar´e exponent xxvii 1.3. Examples xxviii 1.3.1. Schottky products xxviii 1.3.2. Parabolic groups xxix 1.3.3. Geometrically finite and convex-cobounded groups xxix 1.3.4. Counterexamples xxx 1.3.5. R-trees and their isometry groups xxxi 1.4. Patterson–Sullivan theory xxxi 1.4.1. Quasiconformal measures of geometrically finite groups xxxiv 1.5. Appendices xxxv Part 1. Preliminaries 1 Chapter 2. Algebraic hyperbolic spaces 3 2.1. The definition 3 2.2. The hyperboloid model 4 2.3. Isometries of algebraic hyperbolic spaces 7 2.4. Totally geodesic subsets of algebraic hyperbolic spaces 12 2.5. Other models of hyperbolic geometry 15 2.5.1. The (Klein) ball model 16 2.5.2. The half-space model 16 2.5.3. Transitivity of the action of Isom(H)on∂H 18 Chapter 3. R-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces 19 3.1. Graphs and R-trees 19 3.2. CAT(-1) spaces 22 3.2.1. Examples of CAT(-1) spaces 23 3.3. Gromov hyperbolic metric spaces 24 vii viii CONTENTS 3.3.1. Examples of Gromov hyperbolic metric spaces 26 3.4. The boundary of a hyperbolic metric space 27 3.4.1. Extending the Gromov product to the boundary 29 3.4.2. A topology on bord X 32 3.5. The Gromov product in algebraic hyperbolic spaces 35 3.5.1. The Gromov boundary of an algebraic hyperbolic space 39 3.6. Metrics and metametrics on bord X 40 3.6.1. General theory of metametrics 40 3.6.2. The visual metametric based at a point w ∈ X 42 3.6.3. The extended visual metric on bord X 43 3.6.4. The visual metametric based at a point ξ ∈ ∂X 45 Chapter 4. More about the geometry of hyperbolic metric spaces 49 4.1. Gromov triples 49 4.2. Derivatives 50 4.2.1. Derivatives of metametrics 50 4.2.2. Derivatives of maps 51 4.2.3. The dynamical derivative 53 4.3. The Rips condition 54 4.4. Geodesics in CAT(-1) spaces 55 4.5. The geometry of shadows 61 4.5.1. Shadows in regularly geodesic hyperbolic metric spaces 61 4.5.2. Shadows in hyperbolic metric spaces 61 4.6. Generalized polar coordinates 66 Chapter 5. Discreteness 69 5.1. Topologies on Isom(X)69 5.2. Discrete groups of isometries 72 5.2.1. Topological discreteness 74 5.2.2. Equivalence in finite dimensions 76 5.2.3. Proper discontinuity 76 5.2.4. Behavior with respect to restrictions 78 5.2.5. Countability of discrete groups 78 Chapter 6. Classification of isometries and semigroups 79 6.1. Classification of isometries 79 6.1.1. More on loxodromic isometries 81 6.1.2. The story for real hyperbolic spaces 82 6.2. Classification of semigroups 82 6.2.1. Elliptic semigroups 83 6.2.2. Parabolic semigroups 83 6.2.3. Loxodromic semigroups 84 6.3. Proof of the Classification Theorem 85 6.4. Discreteness and focal groups 87 Chapter 7. Limit sets 91 7.1. Modes of convergence to the boundary 91 7.2. Limit sets 93 7.3. Cardinality of the limit set 95 7.4. Minimality of the limit set 96 CONTENTS ix 7.5. Convex hulls 99 7.6. Semigroups which act irreducibly on algebraic hyperbolic spaces 102 7.7. Semigroups of compact type 103 Part 2. The Bishop–Jones theorem 107 Chapter 8. The modified Poincar´e exponent 109 8.1. The Poincar´e exponent of a semigroup 109 8.2. The modified Poincar´e exponent of a semigroup 110 Chapter 9. Generalization of the Bishop–Jones theorem 115 9.1. Partition structures 116 9.2. A partition structure on ∂X 120 9.3. Sufficient conditions for Poincar´e regularity 127 Part 3. Examples 131 Chapter 10. Schottky products 133 10.1. Free products 133 10.2. Schottky products 134 10.3. Strongly separated Schottky products 135 10.4. A partition-structure–like structure 142 10.5. Existence of Schottky products 146 Chapter 11. Parabolic groups 149 11.1. Examples of parabolic groups acting on E∞ 149 11.1.1. The Haagerup property and the absence of a Margulis lemma 150 11.1.2. Edelstein examples 151 11.2. The Poincar´e exponent of a finitely generated parabolic group 155 11.2.1. Nilpotent and virtually nilpotent groups 156 11.2.2. A universal lower bound on the Poincar´e exponent 157 11.2.3. Examples with explicit Poincar´e exponents 158 Chapter 12. Geometrically finite and convex-cobounded groups 165 12.1. Some geometric shapes 165 12.1.1. Horoballs 165 12.1.2. Dirichlet domains 167 12.2. Cobounded and convex-cobounded groups 168 12.2.1. Characterizations of convex-coboundedness 170 12.2.2. Consequences of convex-coboundedness 172 12.3. Bounded parabolic points 172 12.4. Geometrically finite groups 176 12.4.1. Characterizations of geometrical finiteness 177 12.4.2. Consequences of geometrical finiteness 182 12.4.3. Examples of geometrically finite groups 186 Chapter 13. Counterexamples 189 13.1. Embedding R-trees into real hyperbolic spaces 189 13.2. Strongly discrete groups with infinite Poincar´e exponent 193 13.3. Moderately discrete groups which are not strongly discrete 193 xCONTENTS 13.4. Poincar´e irregular groups 194 13.5. Miscellaneous counterexamples 198 Chapter 14. R-trees and their isometry groups 199 14.1. Construction of R-trees by the cone method 199 14.2. Graphs with contractible cycles 202 14.3. The nearest-neighbor projection onto a convex set 204 14.4. Constructing R-trees by the stapling method 205 14.5.
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